A Concise Course in Algebraic Topology
J. P. May
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Contents
Introduction
1
Chapter 1. The fundamental group and some of its applications
1. What is algebraic topology?
2. The fundamental group
3. Dependence on the basepoint
4. Homotopy invariance
5. Calculations: π1 (R) = 0 and π1 (S 1 ) = Z
6. The Brouwer fixed point theorem
7. The fundamental theorem of algebra
5
5
6
7
7
8
10
10
Chapter 2. Categorical language and the van Kampen theorem
1. Categories
2. Functors
3. Natural transformations
4. Homotopy categories and homotopy equivalences
5. The fundamental groupoid
6. Limits and colimits
7. The van Kampen theorem
8. Examples of the van Kampen theorem
13
13
13
14
14
15
16
17
19
Chapter 3. Covering spaces
1. The definition of covering spaces
2. The unique path lifting property
3. Coverings of groupoids
4. Group actions and orbit categories
5. The classification of coverings of groupoids
6. The construction of coverings of groupoids
7. The classification of coverings of spaces
8. The construction of coverings of spaces
21
21
22
22
24
25
27
28
30
Chapter 4. Graphs
1. The definition of graphs
2. Edge paths and trees
3. The homotopy types of graphs
4. Covers of graphs and Euler characteristics
5. Applications to groups
35
35
35
36
37
37
Chapter 5. Compactly generated spaces
1. The definition of compactly generated spaces
2. The category of compactly generated spaces
39
39
40
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vi
CONTENTS
Chapter 6. Cofibrations
1. The definition of cofibrations
2. Mapping cylinders and cofibrations
3. Replacing maps by cofibrations
4. A criterion for a map to be a cofibration
5. Cofiber homotopy equivalence
43
43
44
45
45
46
Chapter 7. Fibrations
1. The definition of fibrations
2. Path lifting functions and fibrations
3. Replacing maps by fibrations
4. A criterion for a map to be a fibration
5. Fiber homotopy equivalence
6. Change of fiber
49
49
49
50
51
52
53
Chapter 8. Based cofiber and fiber sequences
1. Based homotopy classes of maps
2. Cones, suspensions, paths, loops
3. Based cofibrations
4. Cofiber sequences
5. Based fibrations
6. Fiber sequences
7. Connections between cofiber and fiber sequences
57
57
57
58
59
61
61
63
Chapter 9. Higher homotopy groups
1. The definition of homotopy groups
2. Long exact sequences associated to pairs
3. Long exact sequences associated to fibrations
4. A few calculations
5. Change of basepoint
6. n-Equivalences, weak equivalences, and a technical lemma
65
65
65
66
66
68
69
Chapter 10. CW complexes
1. The definition and some examples of CW complexes
2. Some constructions on CW complexes
3. HELP and the Whitehead theorem
4. The cellular approximation theorem
5. Approximation of spaces by CW complexes
6. Approximation of pairs by CW pairs
7. Approximation of excisive triads by CW triads
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73
74
75
76
77
78
79
Chapter 11. The homotopy excision and suspension theorems
1. Statement of the homotopy excision theorem
2. The Freudenthal suspension theorem
3. Proof of the homotopy excision theorem
83
83
85
86
Chapter 12. A little homological algebra
1. Chain complexes
2. Maps and homotopies of maps of chain complexes
3. Tensor products of chain complexes
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91
91
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CONTENTS
4. Short and long exact sequences
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93
Chapter 13. Axiomatic and cellular homology theory
1. Axioms for homology
2. Cellular homology
3. Verification of the axioms
4. The cellular chains of products
5. Some examples: T , K, and RP n
95
95
97
100
101
103
Chapter 14. Derivations of properties from the axioms
1. Reduced homology; based versus unbased spaces
2. Cofibrations and the homology of pairs
3. Suspension and the long exact sequence of pairs
4. Axioms for reduced homology
5. Mayer-Vietoris sequences
6. The homology of colimits
107
107
108
109
110
112
114
Chapter 15. The Hurewicz and uniqueness theorems
1. The Hurewicz theorem
2. The uniqueness of the homology of CW complexes
117
117
119
Chapter 16. Singular homology theory
1. The singular chain complex
2. Geometric realization
3. Proofs of the theorems
4. Simplicial objects in algebraic topology
5. Classifying spaces and K(π, n)s
123
123
124
125
126
128
Chapter 17. Some more homological algebra
1. Universal coefficients in homology
2. The Kă
unneth theorem
3. Hom functors and universal coefficients in cohomology
4. Proof of the universal coefficient theorem
5. Relations between ⊗ and Hom
131
131
132
133
135
136
Chapter 18. Axiomatic and cellular cohomology theory
1. Axioms for cohomology
2. Cellular and singular cohomology
3. Cup products in cohomology
4. An example: RP n and the Borsuk-Ulam theorem
5. Obstruction theory
137
137
138
139
140
142
Chapter 19. Derivations of properties from the axioms
1. Reduced cohomology groups and their properties
2. Axioms for reduced cohomology
3. Mayer-Vietoris sequences in cohomology
4. Lim1 and the cohomology of colimits
5. The uniqueness of the cohomology of CW complexes
145
145
146
147
148
149
Chapter 20. The Poincar´e duality theorem
1. Statement of the theorem
151
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viii
CONTENTS
2.
3.
4.
5.
6.
The definition of the cap product
Orientations and fundamental classes
The proof of the vanishing theorem
The proof of the Poincar´e duality theorem
The orientation cover
153
155
158
160
163
Chapter 21. The index of manifolds; manifolds with boundary
1. The Euler characteristic of compact manifolds
2. The index of compact oriented manifolds
3. Manifolds with boundary
4. Poincar´e duality for manifolds with boundary
5. The index of manifolds that are boundaries
165
165
166
168
169
171
Chapter 22. Homology, cohomology, and K(π, n)s
1. K(π, n)s and homology
2. K(π, n)s and cohomology
3. Cup and cap products
4. Postnikov systems
5. Cohomology operations
175
175
177
179
182
184
Chapter 23. Characteristic classes of vector bundles
1. The classification of vector bundles
2. Characteristic classes for vector bundles
3. Stiefel-Whitney classes of manifolds
4. Characteristic numbers of manifolds
5. Thom spaces and the Thom isomorphism theorem
6. The construction of the Stiefel-Whitney classes
7. Chern, Pontryagin, and Euler classes
8. A glimpse at the general theory
187
187
189
191
193
194
196
197
200
Chapter 24. An introduction to K-theory
1. The definition of K-theory
2. The Bott periodicity theorem
3. The splitting principle and the Thom isomorphism
4. The Chern character; almost complex structures on spheres
5. The Adams operations
6. The Hopf invariant one problem and its applications
203
203
206
208
211
213
215
Chapter 25. An introduction to cobordism
1. The cobordism groups of smooth closed manifolds
2. Sketch proof that N∗ is isomorphic to π∗ (T O)
3. Prespectra and the algebra H∗ (T O; Z2 )
4. The Steenrod algebra and its coaction on H∗ (T O)
5. The relationship to Stiefel-Whitney numbers
6. Spectra and the computation of π∗ (T O) = π∗ (M O)
7. An introduction to the stable category
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219
220
223
226
228
230
232
Suggestions for further reading
1. A classic book and historical references
2. Textbooks in algebraic topology and homotopy theory
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235
235
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CONTENTS
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
Books on CW complexes
Differential forms and Morse theory
Equivariant algebraic topology
Category theory and homological algebra
Simplicial sets in algebraic topology
The Serre spectral sequence and Serre class theory
The Eilenberg-Moore spectral sequence
Cohomology operations
Vector bundles
Characteristic classes
K-theory
Hopf algebras; the Steenrod algebra, Adams spectral sequence
Cobordism
Generalized homology theory and stable homotopy theory
Quillen model categories
Localization and completion; rational homotopy theory
Infinite loop space theory
Complex cobordism and stable homotopy theory
Follow-ups to this book
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236
237
237
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238
238
238
239
239
240
240
240
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Introduction
The first year graduate program in mathematics at the University of Chicago
consists of three three-quarter courses, in analysis, algebra, and topology. The first
two quarters of the topology sequence focus on manifold theory and differential
geometry, including differential forms and, usually, a glimpse of de Rham cohomology. The third quarter focuses on algebraic topology. I have been teaching the
third quarter off and on since around 1970. Before that, the topologists, including
me, thought that it would be impossible to squeeze a serious introduction to algebraic topology into a one quarter course, but we were overruled by the analysts
and algebraists, who felt that it was unacceptable for graduate students to obtain
their PhDs without having some contact with algebraic topology.
This raises a conundrum. A large number of students at Chicago go into topology, algebraic and geometric. The introductory course should lay the foundations
for their later work, but it should also be viable as an introduction to the subject
suitable for those going into other branches of mathematics. These notes reflect
my efforts to organize the foundations of algebraic topology in a way that caters
to both pedagogical goals. There are evident defects from both points of view. A
treatment more closely attuned to the needs of algebraic geometers and analysts
ˇ
would include Cech
cohomology on the one hand and de Rham cohomology and
perhaps Morse homology on the other. A treatment more closely attuned to the
needs of algebraic topologists would include spectral sequences and an array of
calculations with them. In the end, the overriding pedagogical goal has been the
introduction of basic ideas and methods of thought.
Our understanding of the foundations of algebraic topology has undergone subtle but serious changes since I began teaching this course. These changes reflect
in part an enormous internal development of algebraic topology over this period,
one which is largely unknown to most other mathematicians, even those working in
such closely related fields as geometric topology and algebraic geometry. Moreover,
this development is poorly reflected in the textbooks that have appeared over this
period.
Let me give a small but technically important example. The study of generalized homology and cohomology theories pervades modern algebraic topology.
These theories satisfy the excision axiom. One constructs most such theories homotopically, by constructing representing objects called spectra, and one must then
prove that excision holds. There is a way to do this in general that is no more difficult than the standard verification for singular homology and cohomology. I find
this proof far more conceptual and illuminating than the standard one even when
specialized to singular homology and cohomology. (It is based on the approximation of excisive triads by weakly equivalent CW triads.) This should by now be a
1
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2
INTRODUCTION
standard approach. However, to the best of my knowledge, there exists no rigorous
exposition of this approach in the literature, at any level.
More centrally, there now exist axiomatic treatments of large swaths of homotopy theory based on Quillen’s theory of closed model categories. While I do not
think that a first course should introduce such abstractions, I do think that the exposition should give emphasis to those features that the axiomatic approach shows
to be fundamental. For example, this is one of the reasons, although by no means
the only one, that I have dealt with cofibrations, fibrations, and weak equivalences
much more thoroughly than is usual in an introductory course.
Some parts of the theory are dealt with quite classically. The theory of fundamental groups and covering spaces is one of the few parts of algebraic topology
that has probably reached definitive form, and it is well treated in many sources.
Nevertheless, this material is far too important to all branches of mathematics to
be omitted from a first course. For variety, I have made more use of the fundamental groupoid than in standard treatments,1 and my use of it has some novel
features. For conceptual interest, I have emphasized different categorical ways of
modeling the topological situation algebraically, and I have taken the opportunity
to introduce some ideas that are central to equivariant algebraic topology.
Poincar´e duality is also too fundamental to omit. There are more elegant ways
to treat this topic than the classical one given here, but I have preferred to give the
theory in a quick and standard fashion that reaches the desired conclusions in an
economical way. Thus here I have not presented the truly modern approach that
applies to generalized homology and cohomology theories.2
The reader is warned that this book is not designed as a textbook, although
it could be used as one in exceptionally strong graduate programs. Even then, it
would be impossible to cover all of the material in detail in a quarter, or even in a
year. There are sections that should be omitted on a first reading and others that
are intended to whet the student’s appetite for further developments. In practice,
when teaching, my lectures are regularly interrupted by (purposeful) digressions,
most often directly prompted by the questions of students. These introduce more
advanced topics that are not part of the formal introductory course: cohomology
operations, characteristic classes, K-theory, cobordism, etc., are often first introduced earlier in the lectures than a linear development of the subject would dictate.
These digressions have been expanded and written up here as sketches without
complete proofs, in a logically coherent order, in the last four chapters. These
are topics that I feel must be introduced in some fashion in any serious graduate
level introduction to algebraic topology. A defect of nearly all existing texts is
that they do not go far enough into the subject to give a feel for really substantial
applications: the reader sees spheres and projective spaces, maybe lens spaces, and
applications accessible with knowledge of the homology and cohomology of such
spaces. That is not enough to give a real feeling for the subject. I am aware that
this treatment suffers the same defect, at least before its sketchy last chapters.
Most chapters end with a set of problems. Most of these ask for computations and applications based on the material in the text, some extend the theory
and introduce further concepts, some ask the reader to furnish or complete proofs
1But see R. Brown’s book cited in §2 of the suggestions for further reading.
2That approach derives Poincar´
e duality as a consequence of Spanier-Whitehead and Atiyah
duality, via the Thom isomorphism for oriented vector bundles.
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INTRODUCTION
3
omitted in the text, and some are essay questions which implicitly ask the reader
to seek answers in other sources. Problems marked ∗ are more difficult or more
peripheral to the main ideas. Most of these problems are included in the weekly
problem sets that are an integral part of the course at Chicago. In fact, doing the
problems is the heart of the course. (There are no exams and no grades; students
are strongly encouraged to work together, and more work is assigned than a student
can reasonably be expected to complete working alone.) The reader is urged to try
most of the problems: this is the way to learn the material. The lectures focus on
the ideas; their assimilation requires more calculational examples and applications
than are included in the text.
I have ended with a brief and idiosyncratic guide to the literature for the reader
interested in going further in algebraic topology.
These notes have evolved over many years, and I claim no originality for most
of the material. In particular, many of the problems, especially in the more classical
chapters, are the same as, or are variants of, problems that appear in other texts.
Perhaps this is unavoidable: interesting problems that are doable at an early stage
of the development are few and far between. I am especially aware of my debts to
earlier texts by Massey, Greenberg and Harper, Dold, and Gray.
I am very grateful to John Greenlees for his careful reading and suggestions,
especially of the last three chapters. I am also grateful to Igor Kriz for his suggestions and for trying out the book at the University of Michigan. By far my greatest
debt, a cumulative one, is to several generations of students, far too numerous to
name. They have caught countless infelicities and outright blunders, and they have
contributed quite a few of the details. You know who you are. Thank you.
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CHAPTER 1
The fundamental group and some of its
applications
We introduce algebraic topology with a quick treatment of standard material about the fundamental groups of spaces, embedded in a geodesic proof of the
Brouwer fixed point theorem and the fundamental theorem of algebra.
1. What is algebraic topology?
A topological space X is a set in which there is a notion of nearness of points.
Precisely, there is given a collection of “open” subsets of X which is closed under
finite intersections and arbitrary unions. It suffices to think of metric spaces. In that
case, the open sets are the arbitrary unions of finite intersections of neighborhoods
Uε (x) = {y|d(x, y) < ε}.
A function p : X −→ Y is continuous if it takes nearby points to nearby points.
Precisely, p−1 (U ) is open if U is open. If X and Y are metric spaces, this means
that, for any x ∈ X and ε > 0, there exists δ > 0 such that p(Uδ (x)) ⊂ Uε (p(x)).
Algebraic topology assigns discrete algebraic invariants to topological spaces
and continuous maps. More narrowly, one wants the algebra to be invariant with
respect to continuous deformations of the topology. Typically, one associates a
group A(X) to a space X and a homomorphism A(p) : A(X) −→ A(Y ) to a map
p : X −→ Y ; one usually writes A(p) = p∗ .
A “homotopy” h : p ≃ q between maps p, q : X −→ Y is a continuous map
h : X × I −→ Y such that h(x, 0) = p(x) and h(x, 1) = q(x), where I is the unit
interval [0, 1]. We usually want p∗ = q∗ if p ≃ q, or some invariance property close
to this.
In oversimplified outline, the way homotopy theory works is roughly this.
(1) One defines some algebraic construction A and proves that it is suitably
homotopy invariant.
(2) One computes A on suitable spaces and maps.
(3) One takes the problem to be solved and deforms it to the point that step
2 can be used to solve it.
The further one goes in the subject, the more elaborate become the constructions A and the more horrendous become the relevant calculational techniques.
This chapter will give a totally self-contained paradigmatic illustration of the basic
philosophy. Our construction A will be the “fundamental group.” We will calculate A on the circle S 1 and on some maps from S 1 to itself. We will then use the
computation to prove the “Brouwer fixed point theorem” and the “fundamental
theorem of algebra.”
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6
THE FUNDAMENTAL GROUP AND SOME OF ITS APPLICATIONS
2. The fundamental group
Let X be a space. Two paths f, g : I −→ X from x to y are equivalent if they
are homotopic through paths from x to y. That is, there must exist a homotopy
h : I × I −→ X such that
h(s, 0) = f (s), h(s, 1) = g(s), h(0, t) = x,
and h(1, t) = y
for all s, t ∈ I. Write [f ] for the equivalence class of f . We say that f is a loop if
f (0) = f (1). Define π1 (X, x) to be the set of equivalence classes of loops that start
and end at x.
For paths f : x → y and g : y → z, define g · f to be the path obtained by
traversing first f and then g, going twice as fast on each:
f (2s)
if 0 ≤ s ≤ 1/2
g(2s − 1) if 1/2 ≤ s ≤ 1.
(g · f )(s) =
Define f −1 to be f traversed the other way around: f −1 (s) = f (1 − s). Define cx to
be the constant loop at x: cx (s) = x. Composition of paths passes to equivalence
classes via [g][f ] = [g ·f ]. It is easy to check that this is well defined. Moreover, after
passage to equivalence classes, this composition becomes associative and unital. It is
easy enough to write down explicit formulas for the relevant homotopies. It is more
illuminating to draw a picture of the domain squares and to indicate schematically
how the homotopies are to behave on it. In the following, we assume given paths
f : x → y,
g : y → z,
h : z → w.
and
h · (g · f ) ≃ (h · g) · f
f
g
h
#
#
## #
## #
##
##
## #
## #
##
##
## #
## #
##
##
## #
## #
##
##
## #
## #
cx
g
f
cw
h
f · cx ≃ f
cy · f ≃ f
f
f
cx
GG
GG
GG
GG
GG
GG
G
cx
cy
f
cx
f
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cy
cy
4. HOMOTOPY INVARIANCE
7
Moreover, [f −1 · f ] = [cx ] and [f · f −1 ] = [cy ]. For the first, we have the following
schematic picture and corresponding formula. In the schematic picture,
ft = f |[0, t]
ft−1 = f −1 |[1 − t, 1].
and
f −1
f
cx
GG
GGG
GG
GG
GG
GG f −1
cf (t)
ft
GG t
GG
GG
GG
GG
GG
G
cx
cx
if 0 ≤ s ≤ t/2
f (2s)
h(s, t) = f (t)
if t/2 ≤ s ≤ 1 − t/2
f (2 − 2s) if 1 − t/2 ≤ s ≤ 1.
We conclude that π1 (X, x) is a group with identity element e = [cx ] and inverse
elements [f ]−1 = [f −1 ]. It is called the fundamental group of X, or the first
homotopy group of X. There are higher homotopy groups πn (X, x) defined in
terms of maps S n −→ X. We will get to them later.
3. Dependence on the basepoint
For a path a : x → y, define γ[a] : π1 (X, x) −→ π1 (X, y) by γ[a][f ] = [a·f ·a−1 ].
It is easy to check that γ[a] depends only on the equivalence class of a and is a
homomorphism of groups. For a path b : y → z, we see that γ[b · a] = γ[b] ◦ γ[a]. It
follows that γ[a] is an isomorphism with inverse γ[a−1 ]. For a path b : y → x, we
have γ[b · a][f ] = [b · a][f ][(b · a)−1 ]. If the group π1 (X, x) happens to be Abelian,
which may or may not be the case, then this is just [f ]. By taking b = (a′ )−1 for
another path a′ : x → y, we see that, when π1 (X, x) is Abelian, γ[a] is independent
of the choice of the path class [a]. Thus, in this case, we have a canonical way to
identify π1 (X, x) with π1 (X, y).
4. Homotopy invariance
For a map p : X −→ Y , define p∗ : π1 (X, x) −→ π1 (Y, p(x)) by p∗ [f ] =
[p ◦ f ], where p ◦ f is the composite of p with the loop f : I −→ X. Clearly
p∗ is a homomorphism. The identity map id : X −→ X induces the identity
homomorphism. For a map q : Y −→ Z, q∗ ◦ p∗ = (q ◦ p)∗ .
Now suppose given two maps p, q : X −→ Y and a homotopy h : p ≃ q. We
would like to conclude that p∗ = q∗ , but this doesn’t quite make sense because
homotopies needn’t respect basepoints. However, the homotopy h determines the
path a : p(x) → q(x) specified by a(t) = h(x, t), and the next best thing happens.
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8
THE FUNDAMENTAL GROUP AND SOME OF ITS APPLICATIONS
Proposition. The following diagram is commutative:
π1 (X, x)
xxx
q
q
xxqx∗
p∗ qq
q
q
xxx
q
q
x8
q
xq
G π1 (Y, q(x)).
π1 (Y, p(x))
γ[a]
Proof. Let f : I −→ X be a loop at x. We must show that q ◦ f is equivalent
to a · (p ◦ f ) · a−1 . It is easy to check that this is equivalent to showing that cp(x) is
equivalent to a−1 · (q ◦ f )−1 · a · (p ◦ f ). Define j : I × I −→ Y by j(s, t) = h(f (s), t).
Then
j(s, 0) = (p ◦ f )(s),
j(s, 1) = (q ◦ f )(s),
and j(0, t) = a(t) = j(1, t).
Note that j(0, 0) = p(x). Schematically, on the boundary of the square, j is
q◦f
G
a
y
y
a
G
p◦f
Thus, going counterclockwise around the boundary starting at (0, 0), we traverse
a−1 · (q ◦ f )−1 · a · (p ◦ f ). The map j induces a homotopy through loops between
this composite and cp(x) . Explicitly, a homotopy k is given by k(s, t) = j(rt (s)),
where rt : I −→ I × I maps successive quarter intervals linearly onto the edges of
the bottom left subsquare of I × I with edges of length t, starting at (0, 0):
o
y
G
5. Calculations: π1 (R) = 0 and π1 (S 1 ) = Z
Our first calculation is rather trivial. We take the origin 0 as a convenient
basepoint for the real line R.
Lemma. π1 (R, 0) = 0.
Proof. Define k : R × I −→ R by k(s, t) = (1 − t)s. Then k is a homotopy
from the identity to the constant map at 0. For a loop f : I −→ R at 0, define
h(s, t) = k(f (s), t). The homotopy h shows that f is equivalent to c0 .
Consider the circle S 1 to be the set of complex numbers x = y + iz of norm 1,
y + z 2 = 1. Observe that S 1 is a group under multiplication of complex numbers.
It is a topological group: multiplication is a continuous function. We take the
identity element 1 as a convenient basepoint for S 1 .
2
Theorem. π1 (S 1 , 1) ∼
= Z.
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5. CALCULATIONS: π1 (R) = 0 AND π1 (S 1 ) = Z
9
Proof. For each integer n, define a loop fn in S 1 by fn (s) = e2πins . This is
the composite of the map I −→ S 1 that sends s to e2πis and the nth power map on
S 1 ; if we identify the boundary points 0 and 1 of I, then the first map induces the
evident identification of I/∂I with S 1 . It is easy to check that [fm ][fn ] = [fm+n ],
and we define a homomorphism i : Z −→ π1 (S 1 , 1) by i(n) = [fn ]. We claim that
i is an isomorphism. The idea of the proof is to use the fact that, locally, S 1 looks
just like R.
Define p : R −→ S 1 by p(s) = e2πis . Observe that p wraps each interval [n, n+1]
around the circle, starting at 1 and going counterclockwise. Since the exponential
function converts addition to multiplication, we easily check that fn = p ◦ f˜n, where
f˜n is the path in R defined by f˜n (s) = sn.
This lifting of paths works generally. For any path f : I −→ S 1 with f (0) = 1,
there is a unique path f˜ : I −→ R such that f˜(0) = 0 and p ◦ f˜ = f . To see
this, observe that the inverse image in R of any small connected neighborhood in
S 1 is a disjoint union of a copy of that neighborhood contained in each interval
(r + n, r + n + 1) for some r ∈ [0, 1). Using the fact that I is compact, we see
that we can subdivide I into finitely many closed subintervals such that f carries
each subinterval into one of these small connected neighborhoods. Now, proceeding
subinterval by subinterval, we obtain the required unique lifting of f by observing
that the lifting on each subinterval is uniquely determined by the lifting of its initial
point.
Define a function j : π1 (S 1 , 1) −→ Z by j[f ] = f˜(1), the endpoint of the lifted
path. This is an integer since p(f˜(1)) = 1. We must show that this integer is
independent of the choice of f in its path class [f ]. In fact, if we have a homotopy
h : f ≃ g through loops at 1, then the homotopy lifts uniquely to a homotopy
˜
˜ 0) = 0 and p ◦ h
˜ = h. The argument is just the same
h : I × I −→ R such that h(0,
˜
as for f : we use the fact that I × I is compact to subdivide it into finitely many
subsquares such that h carries each into a small connected neighborhood in S 1 . We
˜ by proceeding subsquare by subsquare, starting at
then construct the unique lift h
the lower left, say, and proceeding upward one row of squares at a time. By the
uniqueness of lifts of paths, which works just as well for paths with any starting
˜ t) and d(t) = h(1,
˜ t) specify constant paths since h(0, t) = 1 and
point, c(t) = h(0,
h(1, t) = 1 for all t. Clearly c is constant at 0, so, again by the uniqueness of lifts
of paths, we must have
˜ 0)
f˜(s) = h(s,
and
˜ 1).
g˜(s) = h(s,
But then our second constant path d starts at f˜(1) and ends at g˜(1).
Since j[fn ] = n by our explicit formula for f˜n , the composite j ◦ i : Z −→ Z is
the identity. It suffices to check that the function j is one-to-one, since then both i
and j will be one-to-one and onto. Thus suppose that j[f ] = j[g]. This means that
f˜(1) = g˜(1). Therefore g˜−1 · f˜ is a loop at 0 in R. By the lemma, [˜
g −1 · f˜] = [c0 ].
It follows upon application of p∗ that
[g −1 ][f ] = [g −1 · f ] = [c1 ].
Therefore [f ] = [g] and the proof is complete.
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10
THE FUNDAMENTAL GROUP AND SOME OF ITS APPLICATIONS
6. The Brouwer fixed point theorem
Let D2 be the unit disk {y + iz|y 2 + z 2 ≤ 1}. Its boundary is S 1 , and we let
i : S 1 −→ D2 be the inclusion. Exactly as for R, we see that π1 (D2 ) = 0 for any
choice of basepoint.
Proposition. There is no continuous map r : D2 −→ S 1 such that r ◦ i = id.
Proof. If there were such a map r, then the composite homomorphism
π1 (S 1 , 1)
i∗
G π1 (D2 , 1)
r∗
G π1 (S 1 , 1)
would be the identity. Since the identity homomorphism of Z does not factor
through the zero group, this is impossible.
Theorem (Brouwer fixed point theorem). Any continuous map
f : D2 −→ D2
has a fixed point.
Proof. Suppose that f (x) = x for all x. Define r(x) ∈ S 1 to be the intersection with S 1 of the ray that starts at f (x) and passes through x. Certainly r(x) = x
if x ∈ S 1 . By writing an equation for r in terms of f , we see that r is continuous.
This contradicts the proposition.
7. The fundamental theorem of algebra
1
Let ι ∈ π1 (S , 1) be a generator. For a map f : S 1 −→ S 1 , define an integer
deg(f ) by letting the composite
π1 (S 1 , 1)
f∗
G π1 (S 1 , f (1))
γ[a]
G π1 (S 1 , 1)
send ι to deg(f )ι. Here a is any path f (1) → 1; γ[a] is independent of the choice
of [a] since π1 (S 1 , 1) is Abelian. If f ≃ g, then deg(f ) = deg(g) by our homotopy
invariance diagram and this independence of the choice of path. Conversely, our
calculation of π1 (S 1 , 1) implies that if deg(f ) = deg(g), then f ≃ g, but we will not
need that for the moment. It is clear that deg(f ) = 0 if f is the constant map at
some point. It is also clear that if fn (x) = xn , then deg(fn ) = n: we built that fact
into our proof that π1 (S 1 , 1) = Z.
Theorem (Fundamental theorem of algebra). Let
f (x) = xn + c1 xn−1 + · · · + cn−1 x + cn
be a polynomial with complex coefficients ci , where n > 0. Then there is a complex
number x such that f (x) = 0. Therefore there are n such complex numbers (counted
with multiplicities).
Proof. Using f (x)/(x−c) for a root c, we see that the last statement will follow
by induction from the first. We may as well assume that f (x) = 0 for x ∈ S 1 . This
allows us to define fˆ : S 1 −→ S 1 by fˆ(x) = f (x)/|f (x)|. We proceed to calculate
deg(fˆ). Suppose first that f (x) = 0 for all x such that |x| ≤ 1. This allows us to
define h : S 1 × I −→ S 1 by h(x, t) = f (tx)/|f (tx)|. Then h is a homotopy from the
constant map at f (0)/|f (0)| to fˆ, and we conclude that deg(fˆ) = 0. Suppose next
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7. THE FUNDAMENTAL THEOREM OF ALGEBRA
11
that f (x) = 0 for all x such that |x| ≥ 1. This allows us to define j : S 1 × I −→ S 1
by j(x, t) = k(x, t)/|k(x, t)|, where
k(x, t) = tn f (x/t) = xn + t(c1 xn−1 + tc2 xn−2 + · · · + tn−1 cn ).
Then j is a homotopy from fn to fˆ, and we conclude that deg(fˆ) = n. One of our
suppositions had better be false!
It is to be emphasized how technically simple this is, requiring nothing remotely
as deep as complex analysis. Nevertheless, homotopical proofs like this are relatively
recent. Adequate language, elementary as it is, was not developed until the 1930s.
PROBLEMS
(1) Let p be a polynomial function on C which has no root on S 1 . Show that
the number of roots of p(z) = 0 with |z| < 1 is the degree of the map
pˆ : S 1 −→ S 1 specified by pˆ(z) = p(z)/|p(z)|.
(2) Show that any map f : S 1 −→ S 1 such that deg(f ) = 1 has a fixed point.
(3) Let G be a topological group and take its identity element e as its basepoint. Define the pointwise product of loops α and β by (αβ)(t) =
α(t)β(t). Prove that αβ is equivalent to the composition of paths β · α.
Deduce that π1 (G, e) is Abelian.
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CHAPTER 2
Categorical language and the van Kampen
theorem
We introduce categorical language and ideas and use them to prove the van
Kampen theorem. This method of computing fundamental groups illustrates the
general principle that calculations in algebraic topology usually work by piecing
together a few pivotal examples by means of general constructions or procedures.
1. Categories
Algebraic topology concerns mappings from topology to algebra. Category
theory gives us a language to express this. We just record the basic terminology,
without being overly pedantic about it.
A category C consists of a collection of objects, a set C (A, B) of morphisms
(also called maps) between any two objects, an identity morphism idA ∈ C (A, A)
for each object A (usually abbreviated id), and a composition law
◦ : C (B, C) × C (A, B) −→ C (A, C)
for each triple of objects A, B, C. Composition must be associative, and identity
morphisms must behave as their names dictate:
h ◦ (g ◦ f ) = (h ◦ g) ◦ f,
id ◦f = f,
and f ◦ id = f
whenever the specified composites are defined. A category is “small” if it has a set
of objects.
We have the category S of sets and functions, the category U of topological
spaces and continuous functions, the category G of groups and homomorphisms,
the category A b of Abelian groups and homomorphisms, and so on.
2. Functors
A functor F : C −→ D is a map of categories. It assigns an object F (A) of
D to each object A of C and a morphism F (f ) : F (A) −→ F (B) of D to each
morphism f : A −→ B of C in such a way that
F (idA ) = idF (A)
F (g ◦ f ) = F (g) ◦ F (f ).
and
More precisely, this is a covariant functor. A contravariant functor F reverses the
direction of arrows, so that F sends f : A −→ B to F (f ) : F (B) −→ F (A) and
satisfies F (g ◦ f ) = F (f ) ◦ F (g). A category C has an opposite category C op
with the same objects and with C op (A, B) = C (B, A). A contravariant functor
F : C −→ D is just a covariant functor C op −→ D.
For example, we have forgetful functors from spaces to sets and from Abelian
groups to sets, and we have the free Abelian group functor from sets to Abelian
groups.
13
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14
CATEGORICAL LANGUAGE AND THE VAN KAMPEN THEOREM
3. Natural transformations
A natural transformation α : F −→ G between functors C −→ D is a map of
functors. It consists of a morphism αA : F (A) −→ G(A) for each object A of C
such that the following diagram commutes for each morphism f : A −→ B of C :
F (A)
F (f )
αB
αA
G(A)
G F (B)
G(f )
G G(B).
Intuitively, the maps αA are defined in the same way for every A.
For example, if F : S −→ A b is the functor that sends a set to the free
Abelian group that it generates and U : A b −→ S is the forgetful functor that
sends an Abelian group to its underlying set, then we have a natural inclusion of
sets S −→ U F (S). The functors F and U are left adjoint and right adjoint to each
other, in the sense that we have a natural isomorphism
A b(F (S), A) ∼
= S (S, U (A))
for a set S and an Abelian group A. This just expresses the “universal property”
of free objects: a map of sets S −→ U (A) extends uniquely to a homomorphism of
groups F (S) −→ A. Although we won’t bother with a formal definition, the notion
of an adjoint pair of functors will play an important role later on.
Two categories C and D are equivalent if there are functors F : C −→ D and
G : D −→ C and natural isomorphisms F G −→ Id and GF −→ Id, where the Id
are the respective identity functors.
4. Homotopy categories and homotopy equivalences
Let T be the category of spaces X with a chosen basepoint x ∈ X; its morphisms are continuous maps X −→ Y that carry the basepoint of X to the basepoint
of Y . The fundamental group specifies a functor T −→ G , where G is the category
of groups and homomorphisms.
When we have a (suitable) relation of homotopy between maps in a category
C , we define the homotopy category hC to be the category with the same objects
as C but with morphisms the homotopy classes of maps. We have the homotopy
category hU of unbased spaces. On T , we require homotopies to map basepoint to
basepoint at all times t, and we obtain the homotopy category hT of based spaces.
The fundamental group is a homotopy invariant functor on T , in the sense that it
factors through a functor hT −→ G .
A homotopy equivalence in U is an isomorphism in hU . Less mysteriously, a
map f : X −→ Y is a homotopy equivalence if there is a map g : Y −→ X such that
both g ◦ f ≃ id and f ◦ g ≃ id. Working in T , we obtain the analogous notion of
a based homotopy equivalence. Functors carry isomorphisms to isomorphisms, so
we see that a based homotopy equivalence induces an isomorphism of fundamental
groups. The same is true, less obviously, for unbased homotopy equivalences.
Proposition. If f : X −→ Y is a homotopy equivalence, then
f∗ : π1 (X, x) −→ π1 (Y, f (x))
is an isomorphism for all x ∈ X.
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5. THE FUNDAMENTAL GROUPOID
15
Proof. Let g : Y −→ X be a homotopy inverse of f . By our homotopy
invariance diagram, we see that the composites
f∗
g∗
g∗
f∗
π1 (X, x) −→ π1 (Y, f (x)) −→ π1 (X, (g ◦ f )(x))
and
π1 (Y, y) −→ π1 (X, g(y)) −→ π1 (Y, (f ◦ g)(y))
are isomorphisms determined by paths between basepoints given by chosen homotopies g ◦ f ≃ id and f ◦ g ≃ id. Therefore, in each displayed composite, the first
map is a monomorphism and the second is an epimorphism. Taking y = f (x)
in the second composite, we see that the second map in the first composite is an
isomorphism. Therefore so is the first map.
A space X is said to be contractible if it is homotopy equivalent to a point.
Corollary. The fundamental group of a contractible space is zero.
5. The fundamental groupoid
While algebraic topologists often concentrate on connected spaces with chosen
basepoints, it is valuable to have a way of studying fundamental groups that does
not require such choices. For this purpose, we define the “fundamental groupoid”
Π(X) of a space X to be the category whose objects are the points of X and whose
morphisms x −→ y are the equivalence classes of paths from x to y. Thus the set
of endomorphisms of the object x is exactly the fundamental group π1 (X, x).
The term “groupoid” is used for a category all morphisms of which are isomorphisms. The idea is that a group may be viewed as a groupoid with a single object.
Taking morphisms to be functors, we obtain the category G P of groupoids. Then
we may view Π as a functor U −→ G P.
There is a useful notion of a skeleton skC of a category C . This is a “full”
subcategory with one object from each isomorphism class of objects of C , “full”
meaning that the morphisms between two objects of skC are all of the morphisms
between these objects in C . The inclusion functor J : skC −→ C is an equivalence
of categories. An inverse functor F : C −→ skC is obtained by letting F (A)
be the unique object in skC that is isomorphic to A, choosing an isomorphism
αA : A −→ F (A), and defining F (f ) = αB ◦ f ◦ α−1
A : F (A) −→ F (B) for a
morphism f : A −→ B in C . We choose α to be the identity morphism if A is in
skC , and then F J = Id; the αA specify a natural isomorphism α : Id −→ JF .
A category C is said to be connected if any two of its objects can be connected
by a sequence of morphisms. For example, a sequence A ←− B −→ C connects
A to C, although there need be no morphism A −→ C. However, a groupoid C
is connected if and only if any two of its objects are isomorphic. The group of
endomorphisms of any object C is then a skeleton of C . Therefore the previous
paragraph specializes to give the following relationship between the fundamental
group and the fundamental groupoid of a path connected space X.
Proposition. Let X be a path connected space. For each point x ∈ X, the
inclusion π1 (X, x) −→ Π(X) is an equivalence of categories.
Proof. We are regarding π1 (X, x) as a category with a single object x, and it
is a skeleton of Π(X).
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16
CATEGORICAL LANGUAGE AND THE VAN KAMPEN THEOREM
6. Limits and colimits
Let D be a small category and let C be any category. A D-shaped diagram
in C is a functor F : D −→ C . A morphism F −→ F ′ of D-shaped diagrams is a
natural transformation, and we have the category D[C ] of D-shaped diagrams in
C . Any object C of C determines the constant diagram C that sends each object
of D to C and sends each morphism of D to the identity morphism of C.
The colimit, colim F , of a D-shaped diagram F is an object of C together with
a morphism of diagrams ι : F −→ colim F that is initial among all such morphisms.
This means that if η : F −→ A is a morphism of diagrams, then there is a unique
map η˜ : colim F −→ A in C such that η˜ ◦ ι = η. Diagrammatically, this property
is expressed by the assertion that, for each map d : D −→ D′ in D, we have a
commutative diagram
F (d)
G F (D′ )
F (D)
VVtt
t
t Ö
VV ttt
tt Ö
VVι ttt
ttι ÖÖÖ
t
VV t6
ytt ÖÖ
V
Ö
η VV colim F
ÖÖ η
VV
Ö
VV η˜ ƯƯƯ
V' ỊƯƯ
A.
The limit of F is defined by reversing arrows: it is an object lim F of C together
with a morphism of diagrams π : lim F −→ F that is terminal among all such
morphisms. This means that if ε : A −→ F is a morphism of diagrams, then there
is a unique map ε˜ : A −→ lim F in C such that π ◦ ε˜ = ε. Diagrammatically, this
property is expressed by the assertion that, for each map d : D −→ D′ in D, we
have a commutative diagram
F (d)
G F (D′ )
F (D)
TTdrr
uX g
r
u
TT rr
u ØØØ
u
r
u
TTπ rr
uu π Ø
TT r
uu ØØØ
T
Ø
ε TT limy F
ØØ ε
TT
Ø
TT ε˜ ØØØ
T ØØ
A.
If D is a set regarded as a discrete category (only identity morphisms), then
colimits and limits indexed on D are coproducts and products indexed on the set
D. Coproducts are disjoint unions in S or U , wedges (or one-point unions) in T ,
free products in G , and direct sums in A b. Products are Cartesian products in all
of these categories; more precisely, they are Cartesian products of underlying sets,
with additional structure. If D is the category displayed schematically as
GG d′ ,
Gf
eo
or
d
d
where we have displayed all objects and all non-identity morphisms, then the colimits indexed on D are called pushouts or coequalizers, respectively. Similarly, if
D is displayed schematically as
GG d′ ,
Gdo
or
e
f
d
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7. THE VAN KAMPEN THEOREM
17
then the limits indexed on D are called pullbacks or equalizers, respectively.
A given category may or may not have all colimits, and it may have some but
not others. A category is said to be cocomplete if it has all colimits, complete if it
has all limits. The categories S , U , T , G , and A b are complete and cocomplete.
If a category has coproducts and coequalizers, then it is cocomplete, and similarly
for completeness. The proof is a worthwhile exercise.
7. The van Kampen theorem
The following is a modern dress treatment of the van Kampen theorem. I should
admit that, in lecture, it may make more sense not to introduce the fundamental
groupoid and to go directly to the fundamental group statement. The direct proof
is shorter, but not as conceptual. However, as far as I know, the deduction of
the fundamental group version of the van Kampen theorem from the fundamental
groupoid version does not appear in the literature in full generality. The proof well
illustrates how to manipulate colimits formally. We have used the van Kampen
theorem as an excuse to introduce some basic categorical language, and we shall
use that language heavily in our treatment of covering spaces in the next chapter.
Theorem (van Kampen). Let O = {U } be a cover of a space X by path
connected open subsets such that the intersection of finitely many subsets in O is
again in O. Regard O as a category whose morphisms are the inclusions of subsets
and observe that the functor Π, restricted to the spaces and maps in O, gives a
diagram
Π|O : O −→ G P
of groupoids. The groupoid Π(X) is the colimit of this diagram. In symbols,
Π(X) ∼
= colimU∈O Π(U ).
Proof. We must verify the universal property. For a groupoid C and a map
η : Π|O −→ C of O-shaped diagrams of groupoids, we must construct a map
η˜ : Π(X) −→ C of groupoids that restricts to ηU on Π(U ) for each U ∈ O. On
objects, that is on points of X, we must define η˜(x) = ηU (x) for x ∈ U . This is
independent of the choice of U since O is closed under finite intersections. If a path
f : x → y lies entirely in a particular U , then we must define η˜[f ] = η([f ]). Again,
since O is closed under finite intersections, this specification is independent of the
choice of U if f lies entirely in more than one U . Any path f is the composite of
finitely many paths fi , each of which does lie in a single U , and we must define η˜[f ]
to be the composite of the η˜[fi ]. Clearly this specification will give the required
unique map η˜, provided that η˜ so specified is in fact well defined. Thus suppose
that f is equivalent to g. The equivalence is given by a homotopy h : f ≃ g through
paths x → y. We may subdivide the square I × I into subsquares, each of which
is mapped into one of the U . We may choose the subdivision so that the resulting
subdivision of I × {0} refines the subdivision used to decompose f as the composite
of paths fi , and similarly for g and the resulting subdivision of I × {1}. We see
that the relation [f ] = [g] in Π(X) is a consequence of a finite number of relations,
each of which holds in one of the Π(U ). Therefore η˜([f ]) = η˜([g]). This verifies the
universal property and proves the theorem.
The fundamental group version of the van Kampen theorem “follows formally.”
That is, it is an essentially categorical consequence of the version just proved.
Arguments like this are sometimes called proof by categorical nonsense.
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